The unambiguous conclusion of the experiment is that phonon scattering is much stronger in nanoscale beam than in bulk material as is obvious from its 3 orders of magnitude less thermal conductivity. Much of it comes from the strong diffusive surface scattering in the nanoscale beam. The Callaway’s model enables us to study the strength of individual scattering mechanisms.
2.7.1.1 Diffusive surface phonon scattering
Below 20 K, the phonon thermal conductance of the undoped beam displays an approximate T3 power law. In the framework of the Callaway analysis, this suggests a
parameter [( /p= Λ Λ −o) 1]/[( /Λ Λ +o) 1].18 Here, Λo = 1.12(d1d2)
1/2 is the MFP in the
limit of completely diffuse surface scattering for a rectangular beam of sides d1 and d2.
For our beams, Λo is 0.2 µm. Hence, from the phonon MFP obtained at 4 K, p is ≈0.7,
indicating that phonon surface scattering is only partially specular; i.e., on average, each phonon is specularly reflected only 1/(1-p) ~ 3 times before being diffusely scattered. This is comparable to the finding of Tighe et al. and is somewhat surprising. The topology of the surfaces suggests a roughness only of order 4 nm, which is likely induced by the CAIBE (chemical assisted ion beam etching) process. By comparison, the thermal phonon wavelength at our lowest temperature exceeds 10nm. Moreover, polished surfaces of macroscopic crystal used in other experiments have been shown to have a much higher specularity (~0.99). The mechanisms controlling surface
specularity in nanostructures clearly warrant further investigation.
2.7.1.2 Phonon-phonon scattering
At higher temperatures, from 20 to 40K, the conductivity of both the bi-layer and undoped beams falls far below values extrapolated from the T3 power law found below
20K. According to the Callaway analysis, this must arise from a reduction in the MFP corresponding to an increase in scattering. This phenomenon is very common in bulk crystals beyond 10 - 15 K when Umklapp phonon-phonon processes become dominant. In our mesoscopic beams, however, diffuse surface scattering is so strong that the relative contribution due to phonon-phonon scattering remains insignificant until much higher temperatures (> 40K). Surprisingly, the phonon-phonon scattering rate,
deduced from the curve fitting of the Callaway analysis, is approximately 10 times higher than corresponding bulk values.
There are two possible reasons that may account for the high scattering rates that we have deduced. First, the T3 power law obtained from the Callaway analysis is based
on the Debye phonon density of state (DOS) and a constant phonon group velocity. A realistic dispersion relation of GaAs, however, shows that between 1.5 and 2.5 THz, the phonon DOS is higher than the Debye values. Correspondingly, the phonon group velocity is much smaller than that of the low frequency phonons. On the other hand, above 3 THz, the actual phonon DOS is much smaller than the Debye values. The reduction of group velocity reduces the phonon “flow rate” while the reduction of the DOS reduces the density of thermal energy carriers. Both contribute directly to a lower thermal conduction. Taking the realistic dispersion relation into account, a simplified model provides a conductivity value at 40K that is 25% smaller than that obtained using the Debye DOS and constant group velocity. We note that such an argument, while also applicable to bulk crystals, is seldom used. This is because the temperature dependence of the bulk thermal conductivity is dominated by Umklapp
wavelengths approach the scale of surface roughness (< 4nm). In this case, the surface specularity parameter for phonons would become further reduced at higher temperatures. If diffuse surface scattering is assumed to be dominant even at 40K, the average number of specular reflection would decrease to ~1.
Fig. 16 The experimental and Debye phonon density of states of GaAs 0 1 2 3 4 5 6 7 8 0 1 x 1 0 28 2 x 1 0 28 3 x 1 0 28 4 x 1 0 28 5 x 1 0 28 G a A s D e b y e G a A s e xp e rim e n ta l N umber of phonon modes per m 3 per T Hz F re q u e n c y, T H z
2.7.1.3 Phonon electron and defect scattering
The analysis above applies equally well to the bi-layer and undoped beams. However, the data show that from 4 – 40K the bi-layer beams transport heat less efficiently than the undoped beams. This is evident from both the effective thermal conductivity and MFP’s in Fig. 13 and Fig. 15. This appears to indicate that dopants on the top layer of the bi-layer beams introduce additional, efficient mechanisms for phonon scattering. There are three new components that can be identified: scattering of phonons by mobile electrons, by electrons in donor states and by point defects. The latter occurs due to the mass difference between dopants and host atoms. However, the relaxation rates we deduce from the Callaway analysis require an apparent defect density that is ~16 times higher than the known dopant concentration in our heterostructures. Additionally we deduce a phonon-electron scattering rate in our samples that is almost two orders of magnitude higher than that found in macroscopic crystals or obtained from simple models for phonon-electron scattering. The presence of dopants provides extraordinarily strong phonon scattering in nanostructures that requires further investigation.